How to guide on long division of polynomials
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1) The document provides step-by-step instructions for long division with polynomials, dividing the factor x + 1 into the quadratic x^2 - 9x - 10. 2) It works through dividing the leading terms and multiplying the divisor terms to subtract them from the dividend terms. 3) The solution to the division is x - 10, with a remainder of zero, indicating that x + 1 is a factor of the original quadratic x^2 - 9x - 10.
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How to guide on long division of polynomials
- 1. PC - Functions Name __________________ How-to guide to long division with polynomials We are goingto divide the factor 𝑥 + 1 intothe quadratic 𝑥2 − 9𝑥 − 10. Forthe moment,I'll ignore the othertermsand lookjustat the leading 𝑥 of the divisorandthe leading 𝑥2 of the dividend. If I divide the leading 𝑥2 inside bythe leading 𝑥 infront,whatwouldIget? I'd get an 𝑥. So I'll put an 𝑥 ontop: NowI'll take that 𝑥, and multiplyitthroughthe divisor, 𝑥 + 1.First,I multiplythe 𝑥 (ontop) by the 𝑥 (onthe "side"),andcarrythe 𝑥2 underneath. ThenI'll multiplythe 𝑥 (ontop) bythe 1 (onthe "side"),andcarry the 1𝑥 underneath. ThenI'll draw the "equals"bar,so I can do the subtraction. To subtractthe polynomials, I changeall the signs in the secondline... ...andthenI add down.The firstterm(the 𝑥2) will cancel out I needtoremembertocarry downthat last term, the "subtractten",from the dividend. NowI lookat the 𝑥 fromthe divisorandthe newleadingterm, the – 10𝑥,inthe bottomline of the division.If Idivide the – 10𝑥 bythe 𝑥,I wouldendupwitha – 10,so I'll put that on top.
- 2. NowI'll multiplythe – 10 (ontop) bythe leading 𝑥 (onthe "side"),and carrythe – 10𝑥 to the bottom... ...andI'll multiplythe – 10 (ontop) bythe 1 (onthe "side"),andcarry the –10 to the bottom. I draw the equalsbar,and changethesigns on all the termsin the bottomrow ThenI add down. Thenthe solutiontothisdivisionis: 𝒙 – 𝟏𝟎. Since the remainderonthisdivisionwaszero(thatis,since there wasn't anythingleftover),the divisioncame out"even". Whenyoudoregulardivisionwithnumbersandthe divisioncomes out even,itmeansthatthe numberyoudividedbyisa factor of the numberyou're dividing. Forinstance,if you divide 50 by10, the answerwill be a nice whole "5"witha zeroremainder,because 10is a factor of 50. Inthe case of the above polynomial division,the zeroremaindertellsusthat 𝑥 + 1 is a factor of 𝑥2– 9𝑥 – 10, whichyoucan confirm by factoringthe original quadraticdividend, 𝑥2– 9𝑥 – 10.